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Geometry in early years: sowing seeds for a mathematical definition of squares and rectangles

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Abstract

In early years schooling it is becoming common to propose activities that involve moving along paths, or programming robots to do so. In order to promote continuity towards the introduction of geometry in primary school, we developed a long-term teaching experiment (with 15 sessions) carried out over 4 months in a first grade classroom in northern Italy. Students were asked to program a robot to move along paths, to pretend to act as robots and to represent the sequence of commands and the resulting paths. In particular, in this teaching experiment, an overarching mathematical aim was to sow the seeds for a mathematical definition of rectangles that includes squares. Within the paradigm of semiotic mediation, we intended to foster the students’ transition from a dynamic perception of paths to seeing paths also as static wholes, boundaries of figures with sets of geometric characteristics. The students’ situated productions were collected and analysed together with the specific actions of the adults involved, aimed at fostering processes of semiotic mediation. In this paper we analyse the development of the situated texts produced by the students in relation to the pivot signs that were the beginnings of an inclusive definition of rectangles.

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Notes

  1. Visual Gestalts refer to theories of visual perception, developed in Germany in the 1920s, that attempt to describe how people tend to organize visual elements into groups or unified wholes when certain principles are applied.

  2. For a similar process, see for instance the “theory of gears” as reported in Bartolini Bussi et al. (1999)

  3. We want to translate a non-existing Italian word (“quadratizzato”) invented by the students and later used as a pivot sign, so we will use a similar non-existing English word.

  4. Very early on in the activities the students argued that there needed to be four turns because “When she is finished the bee-bot needs to look the same way she started.” This argument was supported through mirroring by S and T, since it was important for the desired evolution of the network of pivot signs around the squarized O.

  5. Not all students were involved in the post-test activity because it was carried out during the final weeks of school, while various end-of-year activities were taking place. In total 7 of the 18 children were selected (2 high achievers, 3 average achievers, 2 low achievers) and assigned the post-test.

  6. This type of response was provided by the two high-achieving students and by one of the average-achieving students.

  7. This type of response was provided by the other two average-achieving students.

  8. This type of response was provided by the two low-achieving students.

  9. This convention is also a design feature in the app Mak-Trace (see Baccaglini-Frank et al. 2014).

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Acknowledgments

We wish to profusely thank: Roberta Munarini for offering to involve her first grade classroom in the project and for her so active collaboration in the project, together with Federica Baroni; the children in the classes and their parents; Alessandro Ramploud for the intercultural issues included in the paper; our Burmese friends, Thein Lwin and Ko Tar, who, thanks to Giuseppe Malpeli, were able to watch and comment on some of the activities reported in this paper, and initiate a programme of international friendship involving Italian and Burmese children.

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Authors and Affiliations

  1. Dipartimento di Educazione e Scienze Umane, Università di Modena e Reggio, Emilia, Italy

    Maria G. Bartolini Bussi & Anna Baccaglini-Frank

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  1. Maria G. Bartolini Bussi
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  2. Anna Baccaglini-Frank
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Correspondence to Maria G. Bartolini Bussi.

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Bartolini Bussi, M.G., Baccaglini-Frank, A. Geometry in early years: sowing seeds for a mathematical definition of squares and rectangles. ZDM Mathematics Education 47, 391–405 (2015). https://doi.org/10.1007/s11858-014-0636-5

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